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(BQ) Part 2 book The finite element method has contents: FEM for plates and shells, FEM for 3D solids, special purpose elements, modelling techniques, FEM for heat transfer problems, using ABAQUS. (BQ) Part 2 book The finite element method has contents: FEM for plates and shells, FEM for 3D solids, special purpose elements, modelling techniques, FEM for heat transfer problems, using ABAQUS.
8 FEM FOR PLATES AND SHELLS 8.1 INTRODUCTION In this chapter, finite element equations for plates and shells are developed The procedure is to first develop FE matrices for plate elements, and the FE matrices for shell elements are then obtained by superimposing the matrices for plate elements and those for 2D solid plane stress elements developed in Chapter Unlike the 2D solid elements in the previous chapter, plate and shell elements are computationally more tedious as they involve more Degrees Of Freedom (DOFs) The constitutive equations may seem daunting to one who may not have a strong background in the mechanics theory of plates and shells, or the integration may be quite involved if it is to be carried out analytically However, the basic concept of formulating the finite element equation always remains the same Readers are advised to pay more attention to the finite element concepts and the procedures outlined in developing plate and shell elements After all, the computer can handle many of the tedious calculations/integrations that are required in the process of forming the elements The basic concepts, procedures and formulations can also be found in many existing textbooks (see, e.g Petyt,1990; Rao, 1999; Zienkiewicz and Taylor, 2000; etc.) 8.2 PLATE ELEMENTS As discussed in Chapter 2, a plate structure is geometrically similar to the structure of the 2D plane stress problem, but it usually carries only transversal loads that lead to bending deformation in the plate For example, consider the horizontal boards on a bookshelf that support the books Those boards can be approximated as a plate structure, and the transversal loads are of course the weight of the books Higher floors of a building are a typical plate structure that carries most of us every day, as are the wings of aircraft, which usually carry loads like the engines, as shown in Figure 2.13 The plate structure can be schematically represented by its middle plane laying on the x–y plane, as shown in Figure 8.1 The deformation caused by the transverse loading on a plate is represented by the deflection and rotation of the normals of the middle plane of the plate, and they will be independent of z and a function of only x and y The element to be developed to model such plate structures is aptly known as the plate element The formulation of a plate element is very much the same as for the 2D solid element, except for the process for deriving the strain matrix in which the theory of plates is used 173 174 CHAPTER FEM FOR PLATES AND SHELLS z, w y Middle plane fz h x Figure 8.1 A plate and its coordinate system Plate elements are normally used to analyse the bending deformation of plate structures and the resulting forces such as shear forces and moments In this aspect, it is similar to the beam element developed in Chapter 5, except that the plate element is two-dimensional whereas the beam element is one-dimensional Like the 2D solid element, a plate element can also be triangular, rectangular or quadrilateral in shape In this book, we cover the development of the rectangular element only, as it is often used Matrices for the triangular element can also be developed easily using similar procedures, and those for the quadrilateral element can be developed using the idea of an isoparametric element discussed for 2D solid elements In fact, the development of a quadrilateral element is much the same as the rectangular element, except for an additional procedure of coordinate mapping, as shown for the case of 2D solid elements There are a number of theories that govern the deformation of plates In this chapter, rectangular elements based on the Mindlin plate theory that works for thick plates will be developed Most books go into great detail to first cover plate elements based on the thin plate theory However, most finite element packages not use plate elements based on thin plate theory In fact, most analysis packages like ABAQUS not even offer the choice of plate elements Instead, one has to use the more general shell elements, also discussed in this chapter Furthermore, using the thin plate theory to develop the finite element equations has a problem, in that the elements developed are usually incompatible or ‘non-conforming’ This means that some components of the rotational displacements may not be continuous on the edges between elements This is because the rotation depends only upon the deflection w in the thin plate theory, and hence the assumed function for w has to be used to calculate the rotation Many texts go into even greater detail to explain the concept, and to prove the conformability of many kinds of thin plate elements To our knowledge, there is really no need, practically, to understand such a concept and proof for readers who are interested in using the finite element method to solve real-life problems In addition, many structures may not be considered as a ‘thin plate’, or rather their transverse shear strains cannot be ignored Therefore, the Reissner–Mindlin plate theory is more suitable in general, and the elements developed based on the Reissner–Mindlin plate theory are more practical and useful This book will only discuss the elements developed based on the Reissner–Mindlin plate theory There are a number of higher order plate theories that can be used for the development of finite elements Since these higher order plate theories are extensions of the 175 8.2 PLATE ELEMENTS Reissner–Mindlin plate theory, there should be no difficulty for readers who can formulate the Mindlin plate element to understand the formulation of higher order plate elements It is assumed that the element has a uniform thickness h If the plate structure has a varying thickness, the structure has to be divided into small elements that can be treated as uniform elements However, the formulation of plate elements with a varying thickness can also be done, as the procedure is similar to that of a uniform element; this would be good homework practice for readers after reading this chapter Consider now a plate that is represented by a two-dimensional domain in the x–y plane, as shown in Figure 8.1 The plate is divided in a proper manner into a number of rectangular elements, as shown in Figure 8.2 Each element will have four nodes and four straight edges At a node, the degrees of freedom include the deflection w, the rotation about x axis θx , and the rotation about y axis θy , making the total DOF of each node three Hence, for a rectangular element with four nodes, the total DOF of the element would be twelve Following the Reissner–Mindlin plate theory (see Chapter 2), its shear deformation will force the cross-section of the plate to rotate in the way shown in Figure 8.3 Any straight fibre that is perpendicular to the middle plane of the plate before the deformation rotates, but remains straight after the deformation The two displacement components that are parallel Figure 8.2 2D domain of a plate meshed by rectangular elements Neutral plane Figure 8.3 Shear deformation in a plate A straight fibre that is perpendicular to the middle plane of the plate before deformation rotates but remains straight after deformation 176 CHAPTER FEM FOR PLATES AND SHELLS to the middle surface can then be expressed mathematically as u(x, y, z) = zθy (x, y) (8.1) v(x, y, z) = −zθy (x, y) where θx and θy are, respectively, the rotations of the fibre of the plate with respect to the x and y axes The in-plane strains can then be given as ε = −zχ where χ is the curvature, given as (8.2) −∂θy /∂x ∂θx /∂y χ = Lθ = ∂θx /∂x − ∂θy /∂y in which L is the differential operator defined in Chapter 2, and is re-written as ∂ − ∂x ∂ L= ∂y ∂ ∂ − ∂x ∂y (8.3) (8.4) The off-plane shear strain is then given as γ = ξxz ξyz ∂w θy + ∂x = ∂w −θx + ∂y (8.5) Note that Hamilton’s principle uses energy functions for derivation of the equation of motion The potential (strain) energy expression for a thick plate element is Ue = h Ae ε T σ dA dz + h Ae τ T γ dA dz (8.6) The first term on the right-hand side of Eq (8.6) is for the in-plane stresses and strains, whereas the second term is for the transverse stresses and strains τ is the average shear stresses relating to the shear strain in the form τ= τxz G =κ τyz 0 γ = κcs γ G (8.7) where G is the shear modulus, and κ is a constant that is usually taken to be π /12 or 5/6 Substituting Eqs (8.2) and (8.7) into Eq (8.6), the potential (strain) energy becomes Ue = Ae h3 T χ cχ dA + 12 Ae κhγ T cs γ dA (8.8) 177 8.2 PLATE ELEMENTS The kinetic energy of the thick plate is given by Te = Ve ρ(u˙ + v˙ + w˙ ) dV (8.9) which is basically a summation of the contributions of three velocity components in the x, y and z directions of all the particles in the entire domain of the plate Substituting Eq (8.1) into the above equation leads to Te = Ae ρ hw˙ + h3 h θ˙ + θ˙ 12 x 12 y where dA = Ae (dT I d) dA w d = θx θy and ρh I=0 0 ρh3 /12 (8.10) (8.11) 0 ρh /12 (8.12) As we can see from Eq (8.10), the terms related to in-plane displacements are less important for thin plates, since it is proportional to the cubic of the plate thickness 8.2.1 Shape Functions It can be seen from the above analysis of the constitutive equations that the rotations, θx and θy are independent of the deflection w Therefore, when it comes to interpolating the generalized displacements, the deflection and rotations can actually be interpolated separately using independent shape functions Therefore, the procedure of field variable interpolation is the same as that for 2D solid problems, except that there are three instead of two DOFs, for a node For four-node rectangular thick plate elements, the deflection and rotations can be summed as 4 Ni wi , w= i=1 θx = Ni θxi , θy = i=1 Ni θyi (8.13) i=1 where the shape function Ni is the same as the four-node 2D solid element in Chapter 7, i.e Ni = 41 (1 + ξi ξ )(1 + ηi η) (8.14) The element constructed will be a conforming element, meaning that w, θx and θy are continuous on the edges between elements Rewriting Eq (8.13) into a single matrix equation, 178 CHAPTER FEM FOR PLATES AND SHELLS we have h w θx = N de θy (8.15) where de is the (generalized) displacement vector for all the nodes in the element, arranged in the order w1 displacement at node θx1 θy1 w2 displacement at node θx2 θy2 (8.16) de = w3 θx3 displacement at node θy3 w4 θx4 displacement at node θy4 e and the shape function matrix is arranged in the order N1 0 N2 0 N3 0 N2 0 N3 N = N1 0 N2 0 0 N1 node node node 0 N3 N4 0 N4 0 N4 (8.17) node 8.2.2 Element Matrices Once the shape function and nodal variables have been defined, element matrices can then be formulated following the standard procedure given in Chapter for 2D solid elements The only difference is that there are three DOFs at one node for plate elements To obtain the element mass matrix me and the element stiffness matrix ke , we have to use the energy functions given by Eqs (8.8) and (8.9) and Hamilton’s principle Substituting Eq (8.15) into the kinetic energy function, Eq (8.9) gives Te = 21 d˙ eT me d˙ e (8.18) where the mass matrix me is given as me = Ae NT I N dA (8.19) The above integration can be carried out analytically, but it will not be detailed in this book Details can be obtained from Petyt [1990] In practice, we often perform the integration numerically using the Gauss integration scheme, discussed in Chapter 179 8.2 PLATE ELEMENTS To obtain the stiffness matrix ke , we substitute Eq (8.15) into Eq (8.6), from which we obtain h3 I T I (8.20) [B ] cB dA + κh[BO ]T cs BO dA ke = Ae Ae 12 The first term in the above equation represents the strain energy associated with the in-plane stress and strains The strain matrix BI has the form of BI = BI1 BI2 BI3 BI4 (8.21) 0 −∂Nj /∂x BIj = 0 ∂Nj /∂y ∂Nj /∂x −∂Nj /∂y Using the expression for the shape functions in Eq (8.14), we obtain where (8.22) ∂Nj ∂Nj ∂ξ = = ξi (1 + ηi η) ∂x ∂ξ ∂x 4a ∂Nj ∂Nj ∂η = = (1 + ξi ξ )ηi ∂y ∂η ∂y 4b (8.23) In deriving Eq (8.23), the relationship ξ = x/a, η = y/b has been employed The second term in Eq (8.20) relates to the strain energy associated with the off-plane shear stress and strain The strain matrix BO has the form BO = BO BO BO BO (8.24) where Nj ∂Nj /∂x (8.25) ∂Nj /∂y −Nj The integration in the stiffness matrix ke , Eq (8.20) can be evaluated analytically as well Practically, however, the Gauss integration scheme is used to evaluate the integrations numerically Note that when the thickness of the plate is reduced, the element becomes over-stiff, a phenomenon that relates to so-called ‘shear locking’ The simplest and most practical means to solve this problem is to use × Gauss points for the integration of the first term, and use only one Gauss point for the second term in Eq (8.20) As for the force vector, we substitute the interpolation of the generalized displacements, given in Eq (8.15), into the usual equation, as in Eq (3.81): fz fe = (8.26) NT dA Ae BO j = which gives the equivalent nodal force vector for the element If the load is uniformly distributed in the element, fz is constant, and the above equation becomes feT = abfz 0 0 0 0 (8.27) Equation (8.27) implies that the distributed force is divided evenly into four concentrated forces of one quarter of the total force 180 CHAPTER FEM FOR PLATES AND SHELLS 8.2.3 Higher Order Elements For an eight-node rectangular thick plate element, the deflection and rotations can be summed as 8 Ni w i , w= i=1 Ni θxi , θx = i=1 θy = Ni θyi (8.28) i=1 where the shape function Ni is the same as the eight-node 2D solid element given by Eq (7.52) The element constructed will be a conforming element, as w, θx and θy are continuous on the edges between elements The formulation procedure is the same as for the rectangular plate elements 8.3 SHELL ELEMENTS A shell structure carries loads in all directions, and therefore undergoes bending and twisting, as well as in-plane deformation Some common examples would be the dome-like design of the roof of a building with a large volume of space; or a building with special architectural requirements such as a church or mosque; or structures with a special functional requirement such as cylindrical and hemispherical water tanks; or lightweight structures like the fuselage of an aircraft, as shown in Figure 8.4 Shell elements have to be used for modelling such structures The simplest but widely used shell element can be formulated easily by combining the 2D solid element formulated in Chapter and the plate element formulated in the previous section The 2D solid elements handle the membrane or in-plane effects, while the plate elements are used to handle bending or off-plane effects The procedure for developing such an element is very similar to the short cut method used to formulate the frame elements using the truss and beam elements, as discussed in Chapter Of course, the shell element can also be formulated using the usual method of defining shape functions, substituting into the constitutive equations, and thus obtaining the element matrices However, as you might have guessed, it is going to be very tedious Bear in mind, however, that the basic concept of deriving the finite element equation still holds, though we will be introducing a so-called short cut method In this book, the derivation for four-nodal, rectangular shell elements will be outlined using the short cut method Figure 8.4 The fuselage of an aircraft can be considered to be a typical shell structure 8.3 SHELL ELEMENTS 181 Since the plate structure can be treated as a special case of the shell structure, the shell element developed in this section is applicable for modelling plate structures In fact, it is common practice to use a shell element offered in a commercial FE package to analyse plate structures 8.3.1 Elements in Local Coordinate Systems Shell structures are usually curved We assume that the shell structure is divided into shell elements that are flat The curvature of the shell is then followed by changing the orientation of the shell elements in space Therefore, if the curvature of the shell is very large, a fine mesh of elements has to be used This assumption sounds rough, but it is very practical and widely used in engineering practice There are alternatives of more accurately formulated shell elements, but they are used only in academic research and have never been implemented in any commercially available software packages Therefore, this book formulates only flat shell elements Similar to the frame structure, there are six DOFs at a node for a shell element: three translational displacements in the x, y and z directions, and three rotational deformations with respect to the x, y and z axes Figure 8.5 shows the middle plane of a rectangular shell element and the DOFs at the nodes The generalized displacement vector for the element can be written as node de1 de2 node de = (8.29) node de3 de4 node where dei (i = 1, 2, 3, 4) are the displacement vector at node i: displacement in x direction ui displacement in y direction vi displacement in z direction wi dei = rotation about x-axis θxi rotation about y-axis θyi rotation about z-axis θzi z, w (−1, +1) (u4, v4,w4, x4, y4, z4) 2b (1, +1) (u3, v3, w3, x3, y3, z3) 2a (−1, −1) (u1, v1, w1, x1, y1, z1) (1, −1) (u2, v2, w2, x2, y2, z2) Figure 8.5 The middle plane of a rectangular shell element (8.30) 182 CHAPTER FEM FOR PLATES AND SHELLS The stiffness matrix for a 2D solid, rectangular element is used for dealing with the membrane effects of the element, which corresponds to DOFs of u and v The membrane stiffness matrix can thus be expressed in the following form using sub-matrices according to the nodes: node1 m k11 m kem = k21 km 31 m k41 node m k12 m k22 m k32 m k42 node m k13 m k23 m k33 m k43 node m k14 m k24 m k34 m k44 node node node node (8.31) where the superscript m stands for the membrane matrix Each sub-matrix will have a dimension of × 2, since it corresponds to the two DOFs u and v at each node Note again that the matrix above is actually the same as the stiffness matrix of the 2D rectangular, solid element, except it is written in terms of sub-matrices according to the nodes The stiffness matrix for a rectangular plate element is used for the bending effects, corresponding to DOFs of w, and θx , θy The bending stiffness matrix can thus be expressed in the following form using sub-matrices according to the nodes: node b k11 b keb = k21 kb 31 b k41 node b k12 b k22 b k32 b k42 node b k13 b k23 b k33 b k43 node b k14 b k24 b k34 b k44 node node node node (8.32) where the superscript b stands for the bending matrix Each bending sub-matrix has a dimension of × The stiffness matrix for the shell element in the local coordinate system is then formulated by combining Eqs (8.31) and (8.32): node m k11 m k21 ke = m k31 m k 41 0 b k11 0 b k21 0 b k31 0 b k41 node m k12 0 0 m k22 0 0 m k32 0 0 m k42 0 0 b k12 0 b k22 0 b k32 0 b k42 node m k13 0 0 m k23 0 0 m k33 0 0 m k43 0 0 b k13 0 b k23 0 b k33 0 b k43 node m k14 0 0 m k24 0 0 m k34 0 0 m k44 0 0 b k14 0 b k24 0 b k34 0 b k44 0 0 0 0 0 0 node node node node (8.33) The stiffness matrix for a rectangular shell matrix has a dimension of 24 × 24 Note that in Eq (8.33), the components related to the DOF θz , are zeros This is because there is 334 CHAPTER 13 USING ABAQUS© properties are actually defined further down the input file There will not be any error stating that the material ALU is invalid regardless of where the material is defined unless it is not defined at all throughout the input file This is true for all other entries into the input file Let us now look at the data lines The first data line in the ∗ BEAM SECTION basically defines the dimensions of the cross-section (0.025 × 0.04 m) Note that the dimensions here are converted to metres to be consistent with the coordinates of the nodes The second data line basically defines the direction cosines indicating the local beam axis What is given is the default values, and this line can actually be omitted in this case The next entry in the model data would be the material properties definition: *MATERIAL, NAME=ALU *ELASTIC, TYPE=ISOTROPIC 69.E9, 0.33 The material for our example is aluminium, and we name it ALU for short All the properties option block will follow after the ∗ MATERIAL option block, which does not require any data lines by itself The ∗ ELASTIC option defines elastic properties, and TYPE=ISOTROPIC defines the material as an isotropic material, i.e the material properties are the same in all directions The data line for the ∗ ELASTIC option includes the values for the Young’s modulus and Poisson’s ratio Depending upon the type of analysis carried out, or the type of material being defined, other properties may need to be defined For example, if a dynamic analysis is required, then the ∗ DENSITY option would also need to be included; or when the material exhibits viscoelastic behaviour, then the ∗ VISCOELASTIC option would be required At this point, we have almost completed describing the model in the model data What is left are the boundary conditions Note that the boundary conditions can also be defined in the history data What can be defined in the model data is only the zero valued conditions *BOUNDARY 1, 1, 6, There is actually more than one way of defining a ∗ BOUNDARY What is shown is the most direct way The first entry into the data line is the node i.d or the name of the node set, if one is defined In this case, since it is only one single node, there is no need for a node set But many times, a problem might involve a whole set of nodes where the same boundary conditions are applied It would thus be more convenient to group these nodes into a set and referenced in the data line The second entry is the first DOF of the node to be constrained, while the third entry is the last DOF to be constrained In ABAQUS, for displacement DOFs, the number 1, and would represent the translational displacements in the x, y and z-directions, respectively, while the numbers 4, and would represent the rotations about the x, y and z-axes, respectively Of course, depending on the type of element used and the type of analysis carried out, there may be other DOFs represented by other numbers (refer to the ABAQUS manual) For example, if a piezoelastic analysis is carried out using piezoelastic elements, there is an additional DOF (other than the displacement 13.5 DEFINING A FINITE ELEMENT MODEL IN ABAQUS 335 DOFs) number representing the electric potential of the node In this case, all the DOFs from to will be constrained to zero (fourth entry in the data line) Strictly speaking, the DOFs available for the 1D planar, beam element in ABAQUS are only 1, and since the others are considered out of plane displacements Since we constrained all six DOFs, ABAQUS will just give a warning during analysis that the constraints on DOFs 3, and will be ignored, since they not exist in this context There are numerous parameters that can actually be included in the keyword line of the ∗ BOUNDARY option if they are required (refer to the ABAQUS manuals for details) For example, the boundary condition can be made to follow an amplitude curve by including AMPLITUDE = Name of amplitude curve definition in the keyword line ABAQUS also provides for certain standard types of zero-valued boundary conditions For example, the above boundary condition can also be written as *BOUNDARY 1, ENCASTRE The word ENCASTRE used in the data line represents a fully built-in condition, which also means that DOFs 1–6 are constrained to zero Other standard boundary conditions are listed in Table 13.2 After defining the boundary condition, we have now completed what is required for the model data of the input file We now need to define the history data As mentioned, the history data would begin with the ∗ STEP option In this example, we would be required to obtain the displacements of the beam as well as the stress along the beam due to the downward force One step would be sufficient here and the loading will be static: *STEP, PERTURBATION *STATIC The perturbation parameter following the ∗ STEP option basically tells ABAQUS that only a linear response should be considered The ∗ STATIC option specifies that a static analysis Table 13.2 Standard boundary condition types in ABAQUS Boundary condition type Description XSYMM YSYMM ZSYMM ENCASTRE PINNED XASYMM YASYMM ZASYMM Symmetry about a plane x = constant (DOFs 1, 5, = 0) Symmetry about a plane y = constant (DOFs 2, 4, = 0) Symmetry about a plane z = constant (DOFs 3, 4, = 0) Fully clamped (DOFs to = 0) Pinned joint (DOFs 1, 2, = 0) Anti-symmetry about a plane x = constant (DOFs 2, 3, = 0) Anti-symmetry about a plane y = constant (DOFs 1, 3, = 0) Anti-symmetry about a plane z = constant (DOFs 1, 2, = 0) 336 CHAPTER 13 USING ABAQUS© is required The next thing to include will be the loading conditions: *CLOAD 11, 2, -1000 ABAQUS offers many types of loading ∗ CLOAD represents concentrated loading, which is the case for our problem Other types of loading include ∗ DLOAD for distributed loading, ∗ DFLUX for distributed thermal flux in thermal-stress analysis, and ∗ CECHARGE for concentrated electric charge for nodes of piezoelectric elements The first entry in the data line is the node i.d or the name of the node set if defined, the second is the DOF the load is applied to, and the third is the value of the load In our case, since the force is acting downward, it is acting on DOF with a negative sign following the convention in ABAQUS Most loadings can also follow an amplitude curve varying with time by including AMPLITUDE = Name of amplitude curve definition in the keyword line This is especially so if transient, dynamic analysis is required For this problem, there is not much more data to include in the history data other than the output requests The user can request the type of outputs he or she wants by indicating as follows: *NODE PRINT, FREQ=1 U, *NODE FILE, FREQ=1 U, *ELEMENT PRINT, FREQ=1 S, E *ELEMENT FILE, FREQ=1 S, E From what we learned from the finite element method, we can actually deduce that certain output variables are direct nodal variables like displacements, while others like stress and strain are actually determined as a distribution in the element using the shape functions In ABAQUS, this difference is categorized into nodal output variables and element output variables ∗ NODE PRINT outputs the results of the required nodal variables in an ASCII text file (.dat file), while the ∗ NODE FILE ouputs the results in a binary format (.fil file) The binary format can be read by post-processors in which the results can be displayed Similarly, ∗ ELEMENT PRINT outputs element variables in ASCII format, while ∗ ELEMENT FILE outputs them in binary format A list of the different output variables can be obtained in the ABAQUS manuals For our case, U in the data lines for ∗ NODE PRINT and ∗ NODE FILE will output all the components of the nodal displacements S and E represent all the components of stress and strain, respectively So if the analysis is run, there will be altogether three tables: one showing the nodal displacements, one showing the stresses in the elements, and the last one showing the strains in the elements The last thing to now is end the step by including ∗ ENDSTEP If multiple steps are present, this would separate the different steps in the history data 13.5 DEFINING A FINITE ELEMENT MODEL IN ABAQUS Running the analysis So the whole input file defining the problem of the cantilever beam is shown below: *HEADING Model of a cantilever beam with a downward force ** *NODE 1, 11, 2.0 *NGEN 1, 11 ** *ELEMENT, TYPE=B21 1, 1, *ELGEN, ELSET=RECT_BEAM 1, 10 ** *BEAM SECTION, ELSET=RECT_BEAM, SECTION=RECT,MATERIAL=ALU 0.025, 0.040 0., 0., -1.0 ** *MATERIAL, NAME=ALU *ELASTIC, TYPE=ISOTROPIC 69.E9, 0.33 ** *BOUNDARY 1, 1, 6, ** *STEP, PERTURBATION *STATIC ** *CLOAD 11, 2, -1000 ** *NODE PRINT, FREQ=1 U, *NODE FILE, FREQ=1 U, *ELEMENT PRINT, FREQ=1 S, E *ELEMENT FILE, FREQ=1 S, E ** *ENDSTEP 337 CHAPTER 13 USING ABAQUS© 338 ABAQUS input files end with the extension inp So if we call this file beam.inp, we can run this example in ABAQUS using the following command at the command prompt (note that, to-date, ABAQUS is usually run on a unix platform): abaqus job = beam Users can check the full syntax of the ABAQUS execution command in the manuals Results After executing the analysis, there could be several results files generated In ASCII text format would be the beam.dat file ABAQUS ouputs its results in ASCII format in the file ending with the extension dat As mentioned, the binary format would be in the file with the fil extension, and is generally used as input for post-processors The dat file can of course be viewed by any text editor, and it will show lots of numbers associated with the input processing, the analysis steps, and lastly, the requested outputs (∗ NODE PRINT and ∗ EL PRINT) These output data can of course be used for plotting graphs or as inputs to other programming codes, depending on the user Many users would view the results using post-processors like ABAQUS/Post, ABAQUS/Viewer, PATRAN, and so on The choice is entirely up to the preference of the user, and of course, the availability of these post-processors In this book, the results are viewed using PATRAN, and the results are shown below Figure 13.3 shows the deformation plot of the cantilever beam as obtained in PATRAN This plot shows how the cantilever beam deforms under the applied loads The actual beam db–default_viewport–default_group–Entity MSC/PATRAN Version 8.5 07-Feb-0217:29:34 Deform: Static, Step1, Total Time = 0._2: Deformation, Displacements 11 22 33 4 5 6 7 9 10 11 10 11 Y Z X default_Deformation: Max 2.90–01 @Nd11 Figure 13.3 Deformation plot from PATRAN 339 13.6 GENERAL PROCEDURES Stress, S11 on top and bottom surfaces of beam (Pa) 2.85+08 Stress, S11 on bottom beam surface Stress, S11 on top beam surface 1.90+08 9.50+07 –-9.50+07 –-1.90+08 –-2.85+08 0.0 0.40 0.80 1.20 1.60 2.0 x/m Figure 13.4 XY-plot of stress, σxx along the beam displacements of the nodes can also be included in the deformation plot, but if there are many nodes, it makes viewing them on-screen difficult Figure 13.4 shows an XY -plot obtained in PATRAN of the stress, σxx , on the top and bottom surface of the beam The plot clearly shows a tensile stress on the top and a compressive stress at the bottom XY -plots of strain and displacements can be similarly obtained in PATRAN 13.6 GENERAL PROCEDURES How to write the ABAQUS input file of a simple problem of a cantilever beam has been shown in the previous section This chapter will not be sufficient to go through the many keywords that are available in ABAQUS, and therefore the focus will not concentrate on that Readers and users should consult the manuals for more information regarding the keywords This section thus aims to provide a general guide not just to using ABAQUS, but generally to most finite element software From the previous example, it can be seen that certain information must be provided for the software to carry out the analysis This information is required to solve for the finite element problem, and it has been highlighted throughout this book that the information is mainly used to formulate the necessary matrices Of course, there are certain parameters that govern the algorithm and the way in which the equations are solved in the program as well So in this sense, there should not be much difference between different software other than the format in which the information is supplied and the way in which results 340 CHAPTER 13 USING ABAQUS© Node Definitions (*NODE, *NGEN, *NCOPY, *NFIL, *NSET) GEOMETRY DEFINITION Element Definitions (*ELEMENT, *ELGEN, *ELCOPY, *ELSET) *BEAM SECTION,*SHELL SECTION, *SOLID SECTION,*MEMBRANE SECTION, etc ELEMENT PROPERTIES DEFINITION *MATERIAL, *ELASTIC, *DENSITY, *VISCOELASTIC, *DAMPING, etc MATERIAL PROPERTIES DEFINITION *BOUNDARY, *CONTACT INTERFERENCE, *CONTACT PAIR, *INITIAL CONDITIONS, etc BOUNDARY AND INITIAL CONDITIONS *CLOAD, *DLOAD, *DFLUX, *CECHARGE, *DECHARGE, *TEMPERATURE, etc LOADING CONDITIONS *STATIC,*STEADY STATE DYNAMICS, *PIEZOELECTRIC, *MASS DIFFUSION, *HEAT TRANSFER, etc ANALYSIS TYPES *NODE PRINT,*NODE FILE, *EL PRINT, *EL FILE, etc OUTPUT REQUESTS Figure 13.5 General information required by finite element software are presented Figure 13.5 is a summary of the general information that finite element software requires to solve most problems The keywords provided on the left are some of the keywords used in ABAQUS to provide that particular information To summarize, we would first need to define the geometry by defining the nodes and the elements Remember that in the finite element method, the whole domain is discretized to small elements This is generally called meshing Next, we would need to provide some properties for the elements used For example, using 1D beam elements would require one to provide the type of crosssection and the cross-section dimension; or when using 2D plate elements it would require the thickness of the plate to be provided, and so on After that we would need to define the properties of the material or materials being used and associated with the elements We would then need to provide information regarding the boundary and initial conditions the model is under This is necessary for the solver to evaluate the equations Similarly for the loading conditions, which must also be provided unless there is no load on the model like in many analyses involving natural frequencies extraction After all this, the model is 13.6 GENERAL PROCEDURES 341 more or less defined The next step would be to tell the software what type of problem or analysis this is Is the problem a static analysis, or a transient dynamic analysis, or a heat transfer analysis? The software would require this information and the user must provide it with the analysis type Finally, the user can also tell the software what are the results he or she is seeking For example, for most applied mechanics problems, the displacements are the true nodal variables that the solver will compute The software, however, can also compute the stress and strain from interpolation of these nodal displacements automatically, and this can be done by specifying them in the input file REFERENCES ABAQUS user’s manual, volumes I, II and III, version 6.1: Hibbitt, Karlsson & Sorensen, Inc., 2000 ABAQUS keywords manual, version 6.1: Hibbitt, Karlsson & Sorensen, Inc., 2000 ABAQUS theory manual, version 6.1: Hibbitt, Karlsson & Sorenson, Inc., 2000 Argyris, J H., Fried, I and Scharpf, D W., The TET 20 and TEA elements for the matrix displacement method Aero J Vol 72, 618–625, 1968 Barsoum, R S., On the use of isoparametric finite elements in linear fracture mechanics, International Journal for Numerical Methods in Engineering, Vol 10, 25–38, 1976 Barsoum, R S., Triangular quarter point elements as elastic and perfectly elastic crack tip elements, International Journal for Numerical Methods in Engineering, Vol 11, 85–98, 1977 Bathe, K J., Finite Element Procedures, Prentice Hall, Englewood Cliffs, 1996 Belytschko, T., Liu, K L and Moran, B., Nonlinear Finite Elements for Continua and Structures John Wiley & Sons, Ltd, 2000 Bettess, P., Infinite Elements, Penshaw Press, 1992 Cheung, Y K., Finite Strip Method in Structured Analysis, Pergamon Press, 1976 Clough, R.W and Penzien, J., Dynamics of Structures, McGraw-Hill, New York, 1975 Cook, R D., Finite Element Modeling for Stress Analysis, John Wiley & Sons, Inc., 1995 Cook, R D., Concepts and Applications of Finite Element Analysis, 2nd edition, John Wiley & Sons, 1981 Crandall, S H., Engineering Analysis: A Survey of Numerical Procedures, McGraw-Hill, New York, 1956 Crocker, M J., editor, Handbook of Acoustics, Chapter John Wiley & Sons, 1998 Daily, J W and Harleman, D R F., Fluid Dynamics, Addison-Wesley, Reading, Mass., 1966 Eisenberg, M A and Malvern, L E., On finite element integration in natural coordinates Int J Num Meth Eng Vol 7, 574–575, 1973 Finlayson, B A., The Method of Weighted Residuals and Variational Principles, Academic Press, New York, 1972 Finlayson, B A and Scriven, L E., The method of weighted residuals – a review, Applied Mechanics Review, Vol 19, 735–748, 1966 Fung, Y C., Foundations of Solid Mechanics, Prentice-Hall, Englewood Cliffs, 1965 Henshell, R D and Shaw, K G., Crack tip elements are unnecessary, International Journal for Numerical Methods in Engineering, Vol 9, 495–509, 1975 Hilber, H M., Hughes, T J R., and Taylor, R L., Collocation, dissipation and ‘overshoot’ for time integration schemes in structural dynamics, Earthquake Engineering and Structural Dynamics, Vol 6, 99–117, 1978 Hughes, J R T., The Finite Element Method Prentice-Hall International, Inc., 1987 Kardestuncer, H editor-in-chief, Finite Element Handbook, McGraw-Hill, 1987 Kausel, E and Roësset, J M., Semianalytic hyperelment for layered strata, Journal of the Engineering Mechanics Division, Vol 103(4), 569, 1977 342 REFERENCES 343 Liu, G R., Mesh Free Methods: Moving Beyond the Finite Element Method, CRC Press, Boca Raton, 2002 Liu G R., A combined finite element/strip element method for analyzing elastic wave scattering by cracks and inclusions in laminates Computational Mechanics Vol 28, 76–81, 2002 Liu, G R and Achenbach, J D., A strip element method for stress analysis of anisotropic linearly elastic solids ASME J Appl Mech., Vol 61, 270–277, 1994 Liu, G R and Achenbach, J D., A strip element method to analyze wave scattering by cracks in anisotropic laminated plates ASME J Appl Mech., Vol 62, 607–613,1995 Liu, G R., Achenbach, J D., Kim, J O and Li, Z L., A combined finite element method/boundary element method for V(z) curves of anisotropic-layer/substrate configurations Journal of the Acoustical Society of America, Vol 92(5), 2734–2740, 1992 Liu, G R and Quek, S S Jerry, A non-reflecting boundary for analyzing wave propagation using the finite element method Finite Elements in Analysis and Design (in press.) Liu, G R and Quek, S S Jerry, A finite element study of the stress and strain fields of InAs quantum dots embedded in GaAs Semiconductor Science and Technology, Vol 17, 630–643, 2002 Liu, G R and Xi, Z C Elastic Waves in Anisotropic Laminates, CRC Press, 2001 Liu, G R., Xu, Y G and Wu, Z P., Total solution for structural mechanics problems Computer Methods in Applied Mechanics and Engineering Vol 191, 989–1012, 2001 Mindlin, R D., Influence of rotary inertia and shear on flexural motion of isotropic elastic plates, Journal of Applied Mechanics, Vol 18, 31–38, 1951 MSC/Dytran user’s manual, version 4: The MacNeal-Schwndler Corporation, USA, 1997 Murnaghan, F D., Finite Deformation of an Elastic Solid, John Wiley & Sons, 1951 NAFEMS, A Finite Element Primer, Dept of Trade and Industry, 1986 Newmark, N M., A method of computation for structural dynamics, ASCE Journal of Engineering Mechanics Division, Vol 85, 67-94, 1959 Ottosen, N S and Pertersson, H., Introduction to the Finite Element Method, Prentice Hall, New York, 1992 Peterson, L A and Londry K J., Finite-element structural analysis: a new tool for bicycle frame design Bike Tech, Vol 5(2), 1986 Petyt, M., Introduction to Finite Element Vibration Analysis, Cambridge University Press, Cambridge, 1990 Quek, S S., NUS industrial attachment report for session 1997/98: DSO National Laboratories Rao, S.S., The Finite Element in Engineering, 3rd edition, Butterworth-Heinemann, 1999 Reddy, J N., Finite Element Method, John Wiley & Sons Inc., New York, 1993 Reddy, J N., Energy and Variational Methods In Engineering, John Wiley, New York, 1984 Reissner, E., The effect of transverse shear deformation on the bending of elastic plates, Journal of Applied Mechanics, Vol 67, A67–A77, 1945 Segerlind, L J., Applied Finite Element Analysis, 2nd edition, John Wiley & Sons, Inc., 1984 Tassoulas, J L and Kausel, E., Elements for the numerical analysis of wave motion in layered strata Int J Numer Methods Eng., Vol 19, 1005–1032, 1983 Timoshenko, S., Theory of Plates and Shells, McGraw, London, 1940 Timoshenko, S P and Goodier, J N., Theory of Elasticity, 3rd edition, McGraw-Hall, New York, 1970 Washizu, K et al Finite Elements Handbook, Vols and Baitukan, Japan (in Japanese), 1981 Zienkiewicz, O C., The Finite Element Method, 4th edition, McGraw-Hill, London, 1989 Zienkiewicz, O C and Taylor, R L., The Finite Element Method, 5th edition, ButterworthHeinemann, 2000 INDEX ABAQUS input file, 327, 329 calculate eigenvalues of beam, 100 calculate low speed impact on bicycle frame, 122 calculation of 2D heat transfer, 318 calculation of stress distribution in quantum dot structure, 227 cantilever beam with downward force, 337 eigenvalue analysis of micro-motor, 186 static analysis of mico-motor, 165 transient analysis of micro-motor, 193 ABAQUS input syntax rules, 327 Acoustic, 287 Admissible displacement, 37 Area coordinates, 137–138 Axisymmetric elements, 262 loading, 250 mesh, model, 224 solids, 250 Bandwidth, 247 Beam element(s), 90–107, 108 Beam(s), 5, 24, 90 Bending of beams, 24 Bending of plates, 28 Boundary conditions cyclic, 264 essential,13, 18–19, 309 for 2D heat transfer, 308, homogenuous, 19 in abaqus input file, 330, 334 infinite, 245 inhomogenuous, 19 natural, 13, 18–19 symmetric, 258, 274 Boundary element method (BEM), 242, 324 Central difference algorithm, 60 Comment lines, 327 Compatibility equations, 37 of displacements, 243 of mesh, 254, 274 Complexity of linear algebraic system, 247 Computational modelling, Conforming element, 177, 180 Consistency, 44–46, 65 Constant strain element, 204 Constant stress element, 204 Constitutive equations beam, 26 plate, 30 one-dimensional solid, 24 three-dimensional solid, 16 two-dimensional solid, 21 Constraints, 57, 330 Continuity, 36, 254–255 Convective boundary condition, 310 Convergence, 78–79, 100 Coordinate mapping, 142, 148–149 Coordinate transformation, 55, 71, 73–74, 112–113, 116–117 CPU time, 247 Crack-like behaviour, 254 Crack tip elements, 233–234 Cubic element(s), 88, 160, 218–219, 222 345 346 INDEX Damping, 242 coefficients, 60 Data lines, 325, 328 Delta function property, 46–52, 64 Direct assembly, 82, 84, 294 Direct integration method, 58, 60, 195–196 Discrete system equations, 8, 38 Discretization, 4, 38–39 Displacement constraints, 57 Displacement interpolation, 39 Displacement method, 41 Dynamic equilibrium equation, 17–18, 24, 28, 32 Effort to accuracy ratio, 246 Eigenvalue analysis, 58–60, 102, 186 Element distortion, 250–252 Element force vector, 55, 307, 308 Element mass matrix, 111, 140, 146–147, 152, 178 Element matrices 2D solid element, 140–141, 145, 152 3D solid element, 204–208, 213–215 beam element, 93–94 frame element, 109–112, 115–116 heat transfer element, 294, 296–297, 300, 302–303, 308 plate element, 178 shell element, 180–183 truss element, 71 Element stiffness matrix, 53, 71, 140, 145, 152, 178–179, 230, 251, 299, 305 Elements with curved edges, 160, 223 Elements with curved surfaces, 222 Euler–Bernoulli beam theory, 25, 121 Field problems, 282 Finite difference method, 8, 36, 60 Fin one-dimensional, 284, 289 two-dimensional, 282–283 Finite element method (FEM) equations, 67, 90, 109, 114 procedure, 38, 64 Finite strip elements, 233, 242 Finite strip method, 243 Finite volume method (FVM), 8, 324 Flexural vibration modes, 185–186, 188 Fluid flow, 287 Force boundary condition, 19 Fourier series, 195–196 superimposition, 263 Frame element, 108 Frame structure, 108, 112 Free vibration, 58 Functional, 37, 54, 279 Galerkin method, 289 Gauss elimination, 9, 58 Gauss integration, 145, 161, 213 Gauss points, 145–146, 161 Gauss’s divergence theorem, 303 Geometry modelling, 248 Global coordinate system, 55–56, 71–74, 112–114, 116–117, 184 h-adaptivity, 79 Hamilton’s principle, 37–39 Heat insulation boundary, 310 Heat source/sink, 314–315 Heat transfer 1D problem, 289 2D problem, 303 across a composite wall, 285, 298 in a long two-dimensional body, 283 in a one-dimensional fin, 284, 289, 296 in a two-dimensional fin, 282 Helmholtz equation, 282 Hexahedron element, 209 Higher order elements brick, 218 one-dimensional, 87 plate, 180 rectangular, 156 tetrahedron, 216 triangular, 153 History data, 329, 331 Homogeneous equation, 58 Implicit method, 63 Infinite domains, 233, 236 INDEX Infinite element, 237 Initial conditions, 4, 61, 246, 330 Integration by parts, 290 Irrotational fluid flow, 287 Isoparametric element, 152 Jacobian matrix, 151–152, 207, 212–213 Joints modelling, 271 Keyword lines, 326, 328 Kinetic energy, 37, 53, 177 Lagrange interpolants, 87 Lagrange multiplier method, 279 Lagrange type elements, 156, 218 Lagragian functional, 37, 54 Lamb waves, 240 Linear element(s), 70, 130 Linear field reproduction, 47, 50–52, 64–65 Linear independence, 46 Linear quadrilateral elements, 148 Linear rectangular elements, 141 Linear triangular elements, 131 Mass matrix, 53 2D solid element, 140, 146–147, 152 3D solid element, 204–205, 213- 215 beam element, 94 frame element, 111, 116 plate element, 178 shell element, 183 truss element, 71, 74 Matrix inversion, 63 Mechanics for solids and structures, 3, 12 Membrane effects, 182–184 Mesh compatibility, 254–255, 274, 276 density, 250, 276 generation, Mindlin plate, 28, 32–33, 174–175 Modal analysis, 58–59 Model data, 329–330 Moments, 26–28, 30–31 Multi-point constraints (MPC), 267, 271, 273–279 Natural coordinates, 91, 142 Newmark’s method, 63 Nodal interpolation functions, see shape functions Non-conforming, 174 Order of elements, 254 Offsets, 265–269 p-adaptivity, 79 Partitions of unity, 47–49, 52, 64–65 Pascal pyramid, 42 Pascal triangle, 41–42 Penalty method, 279 Planar frame element, 109, 110 Planar truss, 74 Plane strain, 20–22, 129–130 Plane stress, 20–21, 129–130 Plate element(s), 5, 28–29, 173–180 Poisson’s equation, 286, 287 Polynomial basis functions, 43 integrand, 145, 161 interpolation, 39 Potential energy, 37 Quadratic element(s), 130, 157, 216, 219 Quadrilateral element(s), 6, 148–149, 151 Rate of convergence, 79 Rectangular element(s), 141–148, 156–158, 307–308 Reproduction property, 44–45, 50, 65, 78 Serendipity type elements, 157, 219 Shape functions properties, 44, 65 standard procedure for constructing, 41 sufficient requirements, 64 Single point constraint, 51, 258 Singularity point, 234 347 348 INDEX Singularity elements, see crack tip elements Space frame, 108–109, 114–120 Space truss(es), 67 Static analysis, 58 Steady state heat transfer, 282, 289 Straddling elements, 255 Strain displacement relation/relationship(s), 16, 21, 23 Strain energy, 37, 52, 179 Strain matrix, 53, 291 beam element, 93 hexahedron element, 209 linear quadrilateral element, 151 linear rectangular element, 144, 307 linear triangular element, 139, 306 mindlin plate element, 179 tetrahedron element, 200 truss element, 70 Streamline function, 287 Stress intensity factor, 234 Strip element method (SEM), 240, 242, 245 Strong form, 36, 38 Subparametric elements, 153 Subspace iteration, 59 Superparametric elements, 153 Supports modelling, 270 Symmetric positive definite (SPD), 57, 59 Symmetry, 256 axial, 262 cyclic, 264 mirror, 256 repetitive, 264 Tetrahedron element, 200–207, 216–218 Torsional deformation, 282, 286 element, 115 Transformation matrix, 56, 72, 74, 113, 117, 184 Transient analysis, 192 dynamic analysis, 336, 341 response, 60 Triangular element(s), 131–141, 153–156, 289, 305–306 Truss(es), 22–24, 67–89 Two-dimensional (2D) heat transfer, 303, 318 Variational principle(s), 36, 38 Vibration, 12 modes, 59 problems, 240 Visualization, 1, Volume coordinates, 201–202 Wave propagation, 240, 242 Weak form, 19, 36 Weighted residual approach, 305 method, 8, 36 ... conditions 21 , 31, ** ** *NSET, NSET=CENTER 21 , 36, 49, 60, 69, 76, 83, 90, 97, 104, 111, 118, 127 , 149, 169, 188, 20 5, 22 0, 23 3, 24 4, 25 3, 26 0, 26 7, 27 4, 27 5, 27 6, 27 7, 27 8, 27 9, 28 0, 28 1, 28 2, 28 3,... b k21 0 b k31 0 b k41 node m k 12 0 0 m k 22 0 0 m k 32 0 0 m k 42 0 0 b k 12 0 b k 22 0 b k 32 0 b k 42 node m k13 0 0 m k23 0 0 m k33 0 0 m k43 0 0 b k13 0 b k23 0 b k33 0 b k43 node m k14 0 0 m k24... (9.5) V 123 4 V 123 4 V 123 4 The volume coordinate can also be viewed as the ratio between the distance of the point P and point to the plane 23 4: L2 = L1 = dP 23 4 , d1 23 4 L2 = dP −134 , d1 23 4 L3